Fixed point theorems in the Fréchet space C(R+) and functional integral equations on an unbounded interval

The purpose of this paper is to establish fixed point theorems for operators in the Fréchet space of continuous functions on the real half-axis. In our considerations we apply the technique of measures of noncompactness in conjunction with the Tikchonov fixed point theorem. The obtained results are applied in the proof of the solvability of a nonlinear functional integral equation with the initial value. Moreover, we show that solutions of that equation are uniformly globally asymptotically attractive. The results presented in the paper allow to improve existence theorems for integral and functional equations obtained earlier in many research papers.